ASYMPTOTIC ESTIMATION OF SOME RECURRENT SEQUENCES

Abstract

Introduction. The training of modern specialists often requires a symbiosis of
knowledge from different fields. Basic knowledge of mathematical and IT disciplines, as well as their
successful application, is a guarantee of the formation of professional skills in such specialists.
Recurrence relations are extremely important for programming. They are used in algorithm analysis,
approximate calculations, dynamic programming, etc. When considering recurrent sequences, one of
the main problems is its solution, that is, it is necessary to express xn in terms of n. This task is not
always solved. Therefore, the question arises of refining n to find xn depending on how the first term
x0 is chosen.
Originality. The purpose of asymptotic methods is to obtain O-estimates and o-estimates in
cases where it is quite difficult to use the function definition for very large (or very small) values of the
argument. Sometimes it is easier to obtain asymptotic information than any other.
Neither the O-score nor the o-score in their usual form are directly applicable for
computational purposes. However, in almost all cases where such estimates are available, it is
possible, after reviewing the proof, to replace the O-estimates with inequalities that contain numerical
constants. For this, at each stage of our actions, we must indicate certain numbers or functions with
certain properties where, when obtaining asymptotic estimates, we limited ourselves to proving the
existence of such numbers or functions. In this work, the result for one class of recurrent infinitesimal recurrent sequences is refined.
Conclusion. In this article, Theorem 1.1 and Theorem 2.1 are formulated and proved, which
are refinements of the results of Theorems 1 and 2.
The conclusions of the theorems have been tested experimentally, which is shown in Tables 1
and 2. For sufficiently large experimental data, they almost coincide with the exact ones, which
indicates the correctness of the theorems